Dátové štruktúry pre externú pamäť

Zdroje:

• Prednaska L07 Erika Demaina z MIT: http://courses.csail.mit.edu/6.851/spring12/lectures/
• Kapitola o B-trees z Cormen, Thomas H., Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms. (ja spom pouzila 3. vydanie)
• onlie algoritmy a paging http://courses.csail.mit.edu/6.854/03/scribe/scribe19.ps

Osnova:

• uvod do externej pamati z prednasky Erika Demaina
  • strucne opakovanie B-trees: struktura, hladanie, vkladanie, odhad vysky
  • strucne opakovanie mergesortu v externej pamati
• uvod do cache-oblivious modelu z prednasky Erika Demaina
  • strucnu uvod do online algoritmov, kompetitivnost, paging
• staticke vyhldavacie stromy (vEB layout) z prednasky Erika Demaina
• nacrt ako zhruba vyzeraju dynamicke stromy (cache oblivious B-trees), nesli sme do vsetkych detailov

External memory model, I/O model

Introduction

• big and slow disk, fast memory of a limited size M words
• disk reads/writes in blocks of size B words
• when analyzing algorithms, count how many disk reads/writes are done

Example: scanning through n elements (e.g. to compute their sum): Theta(ceiling of n/B) memory transfers

B-trees

• I/O cost of binary search tree?
• from CLRS 3rd edition
• parameter T related to block size B (so that each node fits into a constant number of blocks)
• each node v has some number v.n of keys
  • if it is an internal node, it has v.n+1 children
  • keys in a node are sorted
  • each subtree contains only values between two successive keys in the parent
• all leaves are in the same depth
• each node except root has at least T-1 keys and each internal node except root has at least T children
• each node has at most 2T-1 keys and at most 2T children
• height is O(log_T n)

Search:

• O(log_T n) block reads

Insert:

• find a leaf where the new key belongs
• if leaf is full (2T-1 keys), split into two equal parts of size T-1 and insert median to the parent
  • new element can be inserted to one of the new leaves
• we continue splitting ancestors until we find a node which is not full or until we reach the root. If root is full, we create a new root with 2 children (increasing the height of the tree)
• O(log_T n) block reads and writes

Sorting

• mergesort
  • I/O cost of ordinary implementation?
  • create sorted runs of size M using O(N/B) disk reads and writes
  • repeatedly merge M/B-1 runs into 1 - 1 pass uses N/B disk reads and writes
  • log_{M/B-1} (N/M) passes

Summary

• B-trees can do search tree operations (insert, delete, search/predecessor) in Theta(log_{B+1} n) = Theta(log n / log (B+1)) memeory transfers
  • number of memory transfers compared to usual search tree running time: factor of 1/log B
• search optimal in comparison model
  • proof outline through information theory
  • goal of the seach is to tell us between which two elements is query located
  • there are Theta(n) possible answers, thus Theta(logg n) bits of information
  • in normal comparison model one comparison of query to an elements of the set gives us at most 1 bit of information
  • in I/O model reading a block of B elements and comparing query to them gives us in the beste case its position within these B ele,ents, i.e. Theta(log B) bits
  • to get Theta(log n) bits of infomration for the answer, we need Theta(log n/log B) block reads
  • this can be transformed to a more formal and precise proof
• sorting O((N/B) log_{M/B} (N/B)) memory trasfers
  • number of memory transfers compared to usual sorting running time: factor of more than 1/B

Cache oblivious model

Introduction

• in the I/O model the algorithm explicitly requests block transfers, knows B, controls memory allocation
• in the cache oblivious model algorithm does not know B or M, memory operates as a cache
• M/B slots, each holding one block from disk
• algorithm requests reading a word from disk
  • if the block containing this word in cache, no transfer
  • replace one slot with block holding requested item, write original block if needed (1 or 2 transfers)
  • which one to replace: classical on-line problem of paging

Paging

• cache has size k = M/B slots, each holds one page (above called block)
• sequence of page requests, if the requested page not in memory not in memory, bring it in and remove some other page (page fault)
  • goal is to minimize the number of page faults
• optimal offline algorithm (knows the whole sequence of page requests)
  • At a page fault remove the page that will not be used for the longest time
• example of an on-line algoritm: FIFO:
  • at a page fault remove page which is in the memory longest time
  • uses at most k times as many page faults as the optimal algorithm (k-competitive)
  • no deterministic alg. can be better than k-competitive
  • it is conservative: in a segment of requests containing at most k distinct pages it does at most k page faults
  • every conservative alg. is k-competitive
• Compare a conservative paging alg. (e.g. FIFO) on memory with k blocks to optimum offline alg. on a smaller memory of size h - competitive ratio k/(k-h)
  • if h = k/2, we get competitive ratio 2
  • divide input sequence into maximal blocks, each containing k distinct elements (first element of the next block is distinct from the k elements of the previous block)
  • FIFO uses at most k page faults in each block
  • optimum has at most h sets of a block in memory at the beginning - at least k-h pages will cause a fault
  • in fact we can prove k/(k-h+1), works even for k=h

Back to cache-oblivious model

• we analyze algorithms under the assumptiom that paging algorithm uses off-line optimum
• instead it could use e.g. FIFO in memory 2M and increase the number of transfers by a constant factor
• advantages of cache-oblivious model:
  • may adapt to changing M, B
  • good for a whole memory hierarchy (several levels of cache, disk, network,...)
• scanning still Theta(ceiling of N/B)
• search trees still Theta(log_{B+1} N)
• sorting Theta(n/B log_{M/B}(N/B)) but requires big enough M i.e. M=Omega(B^{1+epsilon})

Static cache-oblivious search trees

• first published by Harald Prokop, Master thesis, MIT 1999 [1]
• use perfectly balanced binary search tree
• search algorithm as usual (follows path from root)
• nodes are stored on disk in some order which we can choose
  • the goal is to choose it so that for any B, M and any search we use only few blocks